Daemonic ergotropy of Gaussian quantum states and the role of measurement-induced purification via general-dyne detection

TL;DR

This work derives a general expression for daemonic ergotropy in Gaussian continuous-variable quantum systems, covering both bipartite settings with general-dyne measurements and open systems under continuous monitoring. A key finding is that, for single-mode Gaussian states, ergotropy depends only on energy and purity, making the daemonic ergotropy a function of unconditional energy and conditional purity, and linking daemonic work to measurement-induced purification. The paper identifies optimal general-dyne strategies, showing heterodyne detection is optimal for phase-invariant two-mode squeezed thermal states and detailing steady-state and transient behaviors in continuously monitored oscillators, with zero-temperature purity maximizing unravellings and finite-temperature scenarios favoring specific homodyne or general-dyne choices. These results highlight how correlations and purification govern enhanced work extraction, with practical implications for quantum batteries and energy management in quantum technologies.

Abstract

According to the Maxwell demon paradigm, additional work can be extracted from a classical or quantum system by exploiting information obtained through measurements on a correlated ancillary system. In the quantum setting, the maximum work extractable via unitary operations in such measurement-assisted protocols is referred to as daemonic ergotropy. In this work, we explore this concept in the context of continuous-variable quantum systems, focusing on Gaussian states and general-dyne (Gaussian) measurements. We derive a general expression for the daemonic ergotropy and examine two key scenarios: (i) bipartite Gaussian states where a general-dyne measurement is performed on one of the two parties, and (ii) open Gaussian quantum systems under continuous general-dyne monitoring of the environment. Remarkably, we show that for single-mode Gaussian states, the ergotropy depends solely on the state's energy and purity. This enables us to express the daemonic ergotropy as a simple function of the unconditional energy and the purity of the conditional states, revealing that enhanced daemonic work extraction is directly linked to measurement-induced purification. We illustrate our findings through two paradigmatic examples: extracting daemonic work from a two-mode squeezed thermal state and from a continuously monitored optical parametric oscillator. In both case we identify the optimal general-dyne strategies that maximize the conditional purity and, in turn, the daemonic ergotropy.

Figures (4)

• Figure 1: Pictorial representation of a daemonic work-extraction protocol involving two correlated quantum systems, $S$ and $A$ (here, two quantum harmonic oscillators). A quantum measurement is performed on the auxiliary system $A$, and the resulting information is exploited by a quantum Maxwell's demon to increase the extractable energy from the system $S$.
• Figure 2: (Color online) Steady-state daemonic ergotropy for a continuously monitored optical parametric oscillator interacting with a finite temperature environment ($\nu_{\sf in}=3$) and near criticality ($\tilde{\chi}=0.99$), as a function of the general-dyne parameter $z_{\sf m}$ (green solid line). The red dotted line corresponds to the maximum value obtained by fixing $z_{\sf m,opt}=0.005$, while the blue dashed line corresponds to the daemonic erogtropy obtained via general-dyne detection ($z_{\sf m}=1$).
• Figure 3: (Color online) Daemonic ergotropy for a continuously monitored optical parametric oscillator interacting with a zero temperature environment, and continuously monitored respectively via homodyne detection with $\vartheta_{\sf m}=0$ (red dotted line), homodyne detection with $\vartheta_{\sf m}=\pi/2$ (green solid line) and heterodyne detection (blue dashed line). Parameters are fixed as follows: $\chi/\kappa=0.4$, $n_{th}=0$, $n_0=2$.
• Figure 4: (Color online) Daemonic ergotropy for a continuously monitored optical parametric oscillator interacting with a non-zero temperature environment, and continuously monitored respectively via homodyne detection with $\vartheta_{\sf m}=0$ (red dotted line), homodyne detection with $\vartheta_{\sf m}=\pi/2$ (green solid line) and heterodyne detection (blue dashed line). Parameters are fixed as follows: $\chi/\kappa=0.4$, $n_{th}=1$, $n_0=2$.